(at around 1h 30 mins) In the first two Omen movies, Damien's mark of the Beast was hidden on the top of his scalp. In this movie, it's behind his right ear.
The timeline for The Omen films does not match up. The first film started in 1970 (as evidenced by the dates on the gravestones of Damien's mother) and ended in 1975 when he was 5 years old. The second film states the events of the first film culminated in 1976 (the date of the newspaper Michael is reading in the opening scene), and then moved forward seven years to around 1983, when Damien was about to turn 13. According to the characters' statements, the events of The Final Conflict take place around 1981/82, although Damien is now 32. Since two decades have elapsed since the second film, the film should take place around 2002/03 - albeit this could be owing to the fact that the creators and studio didn't expect the sequels and the corresponding timeline at the time.
(at around 23 mins) The film was shot in the summer, which is obvious by the foliage. At Damien's reception as US Ambassador to the Court of St James, his assistant's wife is shown in the late stages of pregnancy, and it becomes an important plot aspect that her son is born on March 24th. If she were in late pregnancy, it should therefore be late winter, not summer. The foliage does not match the timing in the plot line with her pregnancy and her baby's birth date.
(at around 26 mins) When Damien is in Hyde Park with Kate, her son and dog, they decide to go for lunch. The trees are in full leaf. However, moments later and wearing the same clothes they reach Speakers' Corner and the trees are completely bare of leaves.
(at around 17 mins) The astronomers mention that the stellar alignment heralding the Second Coming will take place in Cassiopeia. However, the coordinates they give - RA 1h 16' 12", Dec +22° - are in Pisces.
(at around 14 mins) The American flag should always be displayed to the far right of any other flags as seen looking forward from the flags. In the Oval Office, the American flag is on the left.
(at around 17 mins) The initial computer simulation of the Second Coming alignment gives the position's declination as -276° 24' 29". Declinations of stars vary between -90° and +90°.
(at around 1h 45 mins) Near the end of the closing credits, there is a note thanking the National Coal Board for their help but the mining town of Swadlincote has been misspelled as 'Swadlingcote'
As stated in the first film, the Daggers of Megiddo must all be used in order to destroy Damien completely. One of them alone would not be enough, and yet the monks each take one of the daggers with them in their separate attempts to kill him.
(at around 18 mins) In the first scene set in the Fernbank Observatory, the rate of acceleration of the stars under observation is given as "...a couple of thousand Parsecs, minimum". In reality, the Parsec is a unit of distance, not acceleration. However, the astronomer who said this might have simply meant the stars had moved "a couple of thousand parsecs" in the time intervals captured on film, thus providing the rate of acceleration.
When Damien sics his hounds on the assassin monk during the hunting sequence, they apparently savage him to death, but none of them have any blood around their mouths (or anywhere else).
(at around 13 mins) When the U.S. Ambassador of Great Britain commits suicide, a dummy head is used.
(at around 30 mins) Damien's introduction for the "World in Focus" show states that he took over Thorn Industries in 1971 and that he has been running it for seven years. However, his earlier conversation with the President established the current year as 1982 not 1978.
(at around 15 mins) Damien states, while talking to the President, that in two years from that time it will be 1984, making it 1982. If that is the case, then The Omen (1976) should be set in the 1950s, not 1976, as it obviously is. This could be due to the fact that at the time, the producers and the studio hadn't anticipated the sequels and the associated timeline.